Jee Math Test Series Quiz-3

The common difference of the A.P. \(b_1, b_2, \ldots, b_m\) is 2 more than the common difference of A.P. \(a_1, a_2, \ldots, a_n\). If \(a_{40}=-159, a_{100}=-399\) and \(b_{100}=a_{70}\), then \(b_1\) is equal to:

If \(3^{2 \sin 2 \alpha-1}, 14\) and \(3^{4-2 \sin 2 \alpha}\) are the first three terms of an A.P. for some \(\alpha\), then the sixth term of this A.P is:

If the sum of the first 20 terms of the series \(\log _{\left(7^{1 / 2}\right)} x+\log _{\left(7^{1 / 3}\right)} x+\log _{\left(7^{1 / 4}\right)} x+\ldots\) is 460, then \(x\) is equal to :

Let \(a_1, a_2, \ldots . ., a_n\) be a given A.P. whose common difference is an integer and \(S_n=a_1+a_2+\ldots .+a_n\). If \(a_1=1, a_n=300\) and \(15 \leq n \leq 50\), then the ordered pair \(\left(S_{n-4}, a_{n-4}\right)\) is equal to :

If the first term of an A.P. is 3 and the sum of its first 25 terms is equal to the sum of its next 15 terms, then the common difference of this A.P. is :

In the sum of the series \(20+19 \frac{3}{5}+19 \frac{1}{5}+18 \frac{4}{5}+\ldots\) upto \(n^{\text {th }}\) term is 488 and then \(n^{\text {th }}\) term is negative, then :

If the sum of first 11 terms of an A.P., \(a_1, a_2, a_3, \ldots\) is \(0\left(a_1 \neq 0\right)\), then the sum of the A.P., \(a_1, a_3, a_5, \ldots, a_{23}\) is \(k a_1\), where \(k\) is equal to :

If the \(10^{\text {th }}\) term of an A.P. is \(\frac{1}{20}\) and its \(20^{\text {th }}\) term is \(\frac{1}{10}\), then the sum of its first 200 terms is:

Let \(f: R \rightarrow R\) be such that for all \(x \in R\left(2^{1+x}+2^{1-x}\right), f(x)\) and \(\left(3^x+3^{-x}\right)\) are in \(A . P\), then the minimum value of \(f(x)\) is:

Five numbers are in A.P., whose sum is 25 and product is 2520 . If one of these five numbers is \(-\frac{1}{2}\), then the greatest number amongst them is:

Let \(\mathrm{S}_{\mathrm{n}}\) denote the sum of the first \(n\) terms of an A.P. If \(\mathrm{S}_4=16\) and \(\mathrm{S}_6=\) -48 , then \(S_{10}\) is equal to:

If \(a_1, a_2, a_3, \ldots . a_n\) are in A.P. and \(a_1+a_4+a_7+\ldots .+a_{16}=114\), then \(a_1+\mathrm{a}_6+\mathrm{a}_{11}+\mathrm{a}_{16}\) is equal to:

Let the sum of the first \(n\) terms of a non-constant A.P., \(a_1, a_2, a_3, \ldots \ldots \ldots \ldots . .\). be \(50 n+\frac{n(n-7)}{2} A\), where \(A\) is a constant. If \(d\) is the common difference of this A.P., then the ordered pair \(\left(d, a_{50}\right)\) is equal to:

If the sum and product of the first three terms in an A.P. are 33 and 1155 , respectively, then a value of its \(11^{\text {th }}\) term is:

The sum of all natural numbers ‘ \(n\) ‘ such that \(100<n<200\) and H.C.F. \((91, n)>1\) is :

If \({ }^n C_4,{ }^n C_5\) and \({ }^n C_6\) are in A.P., then \(n\) can be :

If 19 th term of a non-zero A.P. is zero, then its (49th term) : (29th term) is :

The sum of all two digit positive numbers which when divided by 7 yield 2 or 5 as remainder is:

Let \(\frac{1}{x_1}, \frac{1}{x_2}, \frac{1}{x_3}, \ldots . .,\left(x_i \neq 0\right.\) for \(\left.i=1,2, \ldots ., n\right)\) be in A.P. such that \(x_1=\) 4 and \(x_{21}=20\). If \(n\) is the least positive integer for which \(x_n>50\), then
\(\sum_{i=1}^n\left(\frac{1}{x_i}\right)\) is equal to.

Let \(\mathrm{a}_1, \mathrm{a}_2, \mathrm{a}_3, \ldots, \mathrm{a}_{49}\) be in A.P. such that \(\sum_{\mathrm{k}=0}^{12} \mathrm{a}_{4 \mathrm{k}+1}=416\) and \(\mathrm{a}_9+\mathrm{a}_{43}=66\). If \(\mathrm{a}_1^2+\mathrm{a}_2^2+\ldots+\mathrm{a}_{17}^2=140 \mathrm{~m}\), then \(\mathrm{m}\) is equal to :

For any three positive real numbers \(\mathrm{a}, \mathrm{b}\) and \(\mathrm{c}\), \(9\left(25 a^2+b^2\right)+25\left(c^2-3 a c\right)=15 b(3 a+c)\). Then :

If three positive numbers \(a, b\) and \(c\) are in A.P. such that \(a b c=8\), then the minimum possible value of \(b\) is :

Let \(a_1, a_2, a_3, \ldots ., a_n\), be in A.P. If \(a_3+a_7+a_{11}+a_{15}=72\), then the sum of its first 17 terms is equal to :

Let \(\alpha\) and \(\beta\) be the roots of equation \(p x^2+q x+r=0, p \neq 0\).
If \(\mathrm{p}, \mathrm{q}, \mathrm{r}\) are in A.P and \(\frac{1}{\alpha}+\frac{1}{\beta}=4\), then the value of \(|\alpha-\beta|\) is:

The sum of the first 20 terms common between the series \(3+7+11+\) \(15+\ldots \ldots \ldots\) and \(1+6+11+16+\ldots \ldots\), is

Given sum of the first \(n\) terms of an A.P. is \(2 n+3 n^2\). Another A.P. is formed with the same first term and double of the common difference, the sum of \(n\) terms of the new A.P. is :

In the sum of first \(n\) terms of an A.P. is \(\mathrm{cn}^2\), then the sum of squares of these \(n\) terms is

If the sum of the first \(2 n\) terms of the A.P. \(2,5,8, \ldots\), is equal to the sum of the first \(n\) terms of the A.P. \(57,59,61, \ldots\), then \(n\) equals

If \(f(x+y)=f(x) f(y)\) and \(\sum_{x=1}^{\infty} f(x)=2, x, y \in \mathbf{N}\), where \(\mathbf{N}\) is the set of all natural numbers, then the value of \(\frac{f(4)}{f(2)}\) is :

Let \(a, b, c, d\) and \(p\) be any non-zero distinct real numbers such that \(\left(a^2\right.\) \(\left.+b^2+c^2\right) p^2-2(a b+b c+c d) p+\left(b^2+c^2+d^2\right)=0\). Then :

If \(2^{10}+2^9 \cdot 3^1+2^8 \cdot 3^2+\ldots .+2 \times 3^9+3^{10}=\mathrm{S}-2^{11}\) then \(\mathrm{S}\) is equal to:

If the sum of the second, third and fourth terms of a positive term G.P. is 3 and the sum of its sixth, seventh and eighth terms is 243 , then the sum of the first 50 terms of this G.P. is:

Let \(\alpha\) and \(\beta\) be the roots of \(x^2-3 x+p=0\) and \(\gamma\) and \(\delta\) be the roots of \(x^2-6 x+q=0\). If \(\alpha, \beta, \gamma, \delta\) form a geometric progression. Then ratio \((2 q+p):(2 q-p)\) is

The sum of the first three terms of a G.P. is \(S\) and their product is 27 . Then all such \(S\) lie in :

If \(|x|<1,|y|<1\) and \(x \neq y\), then the sum to infinity of the following series
\(
(x+y)+\left(x^2+x y+y^2\right)+\left(x^3+x^2 y+x y^2+y^3\right)+\ldots . \text { is : }
\)

Let \(S\) be the sum of the first 9 terms of the series :
\(
\{x+k a\}+\left\{x^2+(k+2) a\right\}+\left\{x^3+(k+4) a\right\}+\left\{x^4+(k+6) a\right\}+\ldots \quad \text { where } \quad a \neq 0 \quad \text { and } \quad x \neq 1 . \quad \text { If }
\)
\(
S=\frac{x^{10}-x+45 a(x-1)}{x-1} \text {, then } k \text { is equal to : }
\)

The product \(2^{\frac{1}{4}} \cdot 4^{\frac{1}{16}} \cdot 8^{\frac{1}{48}} \cdot 16^{\frac{1}{128}} \ldots\) to \(\infty\) is equal to:

Let \(a_n\) be the \(n^{\text {th }}\) term of a G.P. of positive terms. If \(\sum_{n=1}^{100} a_{2 n+1}=200\) and \(\sum_{n=1}^{100} a_{2 n}=100\), then \(\sum_{n=1}^{200} a_n\) is equal to :

\(
\text { If } x=\sum_{n=0}^{\infty}(-1)^n \tan ^{2 \mathrm{n}} \theta \text { and } y=\sum_{n=0}^{\infty} \cos ^{2 n} \theta \text {, for } 0<\theta<\frac{\pi}{4} \text {, then : }
\)

The greatest positive integer \(k\), for which \(49^k+1\) is a factor of the sum \(49^{125}+49^{124}+\ldots+49^2+49+1\), is:

Let \(a_1, a_2, a_3, \ldots\) be a G.P. such that \(a_1<0, a_1+a_2=4\) and \(a_3+a_4=16\). If \(\sum_{i=1}^9 a_i=4 \lambda\), then \(\lambda\) is equal to :

The coefficient of \(x^7\) in the expression \((1+x)^{10}+x(1+x)^9+x^2(1+x)^8+\ldots+x^{10}\) is:

If \(\alpha, \beta\) and \(\gamma\) are three consecutive terms of a non-constant G.P. such that the equations \(\alpha x^2+2 \beta x+\gamma=0\) and \(x^2+x-1=0\) have a common root, then \(\alpha(\beta+\gamma)\) is equal to

If three distinct numbers \(a, b, c\) are in G.P. and the equations \(a x^2+2 b x\) \(+c=0\) and \(d x^2+2 e x+f=0\) have a common root, then which one of the following statements is correct?

The product of three consecutive terms of a G.P. is 512. If 4 is added to each of the first and the second of these terms, the three terms now form an A.P. Then the sum of the original three terms of the given G.P. is :

Let \(\alpha\) and \(\beta\) be the roots of the quadratic equation \(x^2 \sin \theta-x(\sin \theta \cos \theta+1)+\cos \theta=0\left(0<\theta<45^{\circ}\right)\), and \(\alpha<\beta\). Then \(\sum_{\mathrm{n}=0}^{\infty}\left(\alpha^{\mathrm{n}}+\frac{(-1)^{\mathrm{n}}}{\beta^{\mathrm{n}}}\right)\) is equal to :

\(
\text { Let } \mathrm{a}_1, \mathrm{a}_2, \ldots, \mathrm{a}_{10} \text { be a G.P. If } \frac{a_3}{a_1}=25 \text {, then } \frac{\mathrm{a}_9}{\mathrm{a}_5} \text { equals : }
\)

Let \(\mathrm{S}_{\mathrm{n}}=1+\mathrm{q}+\mathrm{q}^2+\ldots .+\mathrm{q}^{\mathrm{n}}\) and
\(
\mathrm{T}_{\mathrm{n}}=1+\left(\frac{\mathrm{q}+1}{2}\right)+\left(\frac{\mathrm{q}+1}{2}\right)^2+\ldots+\left(\frac{\mathrm{q}+1}{2}\right)^{\mathrm{n}}
\)
where \(\mathrm{q}\) is a real number and \(\mathrm{q} \neq 1\). If
\({ }^{101} \mathrm{C}_1+{ }^{101} \mathrm{C}_2 \mathrm{~S}_1+\ldots .+{ }^{101} \mathrm{C}_{101} \mathrm{~S}_{100}=\alpha \mathrm{T}_{100}\), then \(\alpha\) is equal to :

If \(a, b\) and \(c\) be three distinct real numbers in G.P. and \(\mathrm{a}+\mathrm{b}+\mathrm{c}=x \mathrm{b}\), then \(x\) cannot be:

If \(b\) is the first term of an infinite \(\mathrm{G}\). \(\mathrm{P}\) whose sum is five, then \(b\) lies in the interval.